Proving $H(a,b) = \hat{H}(a,b)$ using rational zeta series

The outline of Cezar's and my work will be to show (1) $\hat{H}(a,b)$ is equal to a certain rational zeta series, and (2) $H(a,b)$ is equal to that same raitonal zeta series. Thus $H(a,b) = \hat{H}(a,b)$ independent of Don Zagier's proof. Cezar has proven (2) for $b=0$. In this talk, I will prove (1) for all $a,b$.